Spectra and Minimum Distances of Repeat Multiple–Accumulate Codes

In this paper, the ensembles of repeat multiple- accumulate codes (RAm), which are obtained by interconnecting a repeater with a cascade of m accumulate codes through uniform random interleavers, are analyzed. It is proved that the average spectral shapes of these code ensembles are equal to 0 below a threshold distance epsiv
_m
and, moreover, they form a nonincreasing sequence in m converging uniformly to the maximum between the average spectral shape of the linear random ensemble and 0. Consequently the sequence epsiv
_m
converges to the Gilbert-Varshamov (GV) distance. A further analysis allows to conclude that if m ges 2 the RA
^m
are asymptotically good and that epsiv
_m
is the typical normalized minimum distance when the interleaver length goes to infinity. Combining the two results it is possible to conclude that the typical distance of the ensembles RA
^m
converges to the Gilbert-Varshamov bound.